Linear Systems and Projective Models on Algebraic Curves

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IIt holds s_c(2)≦g+ 2 for any algebraic curve C with genus g where. s_c(2) is the minimal degree of plane models of C. Martens and Keem conjectured that s_c(2)=g-t+ 2holds for an algebraic curve C admitting a double covering to a curve of genus t. In this research, we can conclude that Martens and Keem conjecture is true for t=0, 1 and t=2 with g≧10, however we can find an important counter examples for every g≦9. On 05-06, December, 2009, we held 7th Annual Conference of Algebraic Curves and Related Topics at Yokohama National Universiy, on 11-12, December, 2010, we held 8th Annual Conference of Algebraic Curves and Related Topics at Saitama Universiy and on 10-11, December, 2011, we held 9th Annual Conference of Algebraic Curves and Related Topics at Tokyo Metropolitan Universiy, respectively. We can get some good results about the minimal degree for plane models o f an algebraic curve.